and \(\xi _\gamma \rightarrow 1^-\)  as \(\gamma \rightarrow 0^+\) , where \({h_+}(y) = \max {x : h(x) = y}\) .


The final part of the theorem shows that the upper bound is characterised by the quantity \({2}/{(\xi _\gamma ^2\Delta _{}^2(i,\gamma ))}\) . By continuity, we obtain the regret bound \({2}/{(\Delta _{}^2(i,\gamma ))}\) . Furthermore, we note that \( {i} \rightarrow -\mu _i - (-\mu _1) = \mu _1 - \mu _i\)  as \(\gamma \rightarrow 0^+\) , and thus the upper bound simplifies to \(2/{(\mu _1-\mu _i)}^2\) . This agrees with our intuition since \(_\gamma (i) = -\mu _i + (\gamma /2)\sigma _i^2 \rightarrow -\mu _i\)  as \(\gamma \rightarrow 0^+\) , implying that we are in the risk-neutral setting. Thus, the results correspond to those derived for mean-variance bandits [1]} and CVaR bandits [2]}.

Next, we establish an instance-dependent lower bound for the expected pulls of non-optimal arms under consistent algorithms. Consider a class \(C\)  of distributions and define \(S_i = \lbrace \nu ^{\prime }(i) \in C : (\nu ^{\prime }(i)) < (1)\rbrace \) . Define for each non-optimal arm \(i\) ,
\(\eta (i,\gamma ) = \inf _{\nu ^{\prime }(i) \in S_i} \lbrace (\nu (i),\nu ^{\prime }(i))\rbrace ,\) 
